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Full text of "Mathematical And Physical Papers - Iii"

i|r is insensible, reduces itself to npT\ Referring to (12), we see that, in order that ^ may be insensible, it is necessary to suppose, either that q is incomparably greater than n, or that n is incomparably greater than q. On the former supposition the formula (11) gives V==\/!cf which is equivalent to Newton's result. On the latter supposition we get V= Jk (1 + a/3), which is equivalent to Laplace's result.
The reason why sound would be so rapidly stifled were q and n comparable with each other, may be easily seen on taking a common-sense view of the subject. Conceive a mass of air contained in a cylinder in which an air-tight piston fits, which is capable of moving without friction, and which has its outer face exposed to a constant atmospheric pressure; and suppose the air alternately compressed and rarefied by the motion of the piston. If the motion take place with extreme slowness, there will be no sensible change of temperature, and therefore the work done on the air during compression will be given out again by the air during expansion, inasmuch as the pressure on the piston will be the same when the piston is at the same point of the cylinder, whether it be moving forwards or backwards. Similarly, the work done in rarefying the air will be given out again by the atmosphere as the piston returns towards its position of equilibrium, so that the motion would go on without any permanent consumption of labouring force. Next, suppose the motion of the piston somewhat quicker, so that there is a sensible change of temperature produced by condensation and rarefaction. As the piston moves forward in condensing the air, the temperature rises, and therefore the piston has to work against a pressure greater than if there had been no variation of temperature. By the time the piston returns, a good portion of the heat developed by compression has passed off, and therefore the piston is not helped as much in its backward motion by the pressure of the air in the cylinder as it had been opposed in its forward motion. Similarly, as the piston continues its backward motion, rarefying the air, the temperature falls, the pressure of the air in the cylinder is diminished more than corresponds merely to the change of density, and therefore the piston is less helped in opposing the atmospheric pressure than it would have been had the temperature remained constant. But by the time the piston is returning towards its position of